The Copenhagen problem with a quasi-homogeneous potential

Fractal basins of convergence of libration points in the planar Copenhagen problem with a repulsive quasi-homogeneous Manev-type potential

“…According to [11], the expression of the time-independent effective potential in a synodic coordinates system Oxyz is…”

“…In this study, the Copenhagen case is considered as a special case where ∆ = M (Λ + 2eΛ 1 ), with M = 2, Λ = 1, and Λ 1 = 1. Since ∆ must be positive, the parameter e must satisfy the condition e > −0.5 (more details on the involved parameters and their meaning can be found in [11]). Using the transformation from the inertial to the synodic coordinate system and scaling the physical quantities, where we have considered the constant angular velocity of the primaries equal to unity, the equations of motion of the test particle m in the rotating frame of reference arë…”

“…This subsection is devoted to the case when e ∈ (−0.19526, −0.173395), where eleven libration points exist: five on the x−axis (L 1,2,4,6,7 ) in which L 4 is the central libration point, two on y−axis (L 8,9 ) and four on (x, y) plane (L 10,11,12,13 ). The evolution of the basins of convergence, using Newton-Raphson iterative scheme, for three values of parameter e are illustrated in Fig.…”

“…Moreover, the butterfly wing shaped domains of convergence associated with the libration points L 8,9 shown in purple and crimson color exist. The most notable change due to existence of eight extra libration points in the presence of a negative value of the parameter is the following four lobe appear, corresponding to non collinear libration points (L 10,11,12,13 ) in the vicinity of the primaries as well as at the back of both the exotic bugs. Let us denote the region at the back of exotic bugs corresponding to the libration points L 1 (yellow) and L 7 (cyan) by R 1 and R 7 respectively.…”

Orbital analysis in the planar circular Copenhagen problem using polar coordinates

“…The Copenhagen problem has been studied by numerous researchers over the past decades. A large variety of works is devoted on several aspects of the Copenhagen problem such as the crash test in the RTBP, oblate primaries, radiating primaries, chaotic scattering in the Copenhagen problem, the fractal convergence basins, the case of primaries that act as magnetic dipoles, RTBP with quasi‐homogeneous potential, and oblate primaries and out‐of‐plane points of equilibrium …”